\(\mathbf{F} = \langle \mathbf{F}_x(x,y), \mathbf{F}_y(x,y) \rangle\)
Divergence
The divergence of a vector field \(\mathbf{F}\) in \(\mathbb{R}^2\) is defined by: \[\text{div} \, \mathbf{F} = \frac{\partial \mathbf{F}_x}{\partial x} + \frac{\partial \mathbf{F}_y}{\partial y}\] Indicates how much the vector field spreads out or converges at a point.
Curl
The curl of a vector field \(\mathbf{F}\) in \(\mathbb{R}^2\) is defined by: \[\text{curl} \, \mathbf{F} = \frac{\partial \mathbf{F}_y}{\partial x} - \frac{\partial \mathbf{F}_x}{\partial y}\] Measures the rotation or swirling strength at a point.
Laplace Operator
The Laplacian of the vector field \(\mathbf{F}\) is given by: \[\Delta \mathbf{F} = \frac{\partial^2 \mathbf{F}_x}{\partial x^2} + \frac{\partial^2 \mathbf{F}_x}{\partial y^2} + \frac{\partial^2 \mathbf{F}_y}{\partial x^2} + \frac{\partial^2 \mathbf{F}_y}{\partial y^2}\] Describes how the vector field spreads out or compresses at different points.
Directional Derivative
The directional derivative of a vector field measures the rate of change of the field in a specified direction. It is given by: \[D_{\mathbf{v}} \mathbf{F} = \frac{\partial \mathbf{F}_x}{\partial x} v_x + \frac{\partial \mathbf{F}_x}{\partial y} v_y + \frac{\partial \mathbf{F}_y}{\partial x} v_x + \frac{\partial \mathbf{F}_y}{\partial y} v_y\] Represents the rate of change of each component of the field in a given direction.